Simple problem

Given x(0)=-2, y(0)=1, x'(t)=2x(t), y'(t)=-2y(t), solve for x(t) and y(t)

I plugged this into mathematica and got -2e^(2t)=x(t) and e^(-2t) = y(t)

I notice that x'(t) = 2x(t) means that x(t) has to be e^t and there must be a coefficient (2) in front of the t because the only derivative that has this property is e^x. And the coefficient in front of e will be given by x(0). But is there a "method" that I can go about doing this? or can you only do it by observation?